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On a problem of K. Mahler: Diophantine approximation and Cantor sets

Levesley, J., Salp, C. and Velani, S.L. (2006) On a problem of K. Mahler: Diophantine approximation and Cantor sets. Mathematischen Annalen, 338 (1). pp. 97-118. ISSN 1432-1807

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Abstract

Let K denote the middle third Cantor set and . Given a real, positive function ψ let denote the set of real numbers x in the unit interval for which there exist infinitely many such that |x − p/q| < ψ(q). The analogue of the Hausdorff measure version of the Duffin–Schaeffer conjecture is established for . One of the consequences of this is that there exist very well approximable numbers, other than Liouville numbers, in K—an assertion attributed to K. Mahler. Explicit examples of irrational numbers satisfying Mahler’s assertion are also given. Mathematics Subject Classification (2000) Primary 11J83 - Secondary 11J82 - Secondary 11K55

Dedicated to Maurice Dodson on his retirement—finally!

Item Type: Article
Institution: The University of York
Academic Units: The University of York > Mathematics (York)
Depositing User: York RAE Import
Date Deposited: 27 Mar 2009 12:36
Last Modified: 27 Mar 2009 12:36
Published Version: http://dx.doi.org/10.1007/s00208-006-0069-8
Status: Published
Publisher: Springer Verlag (Germany)
Identification Number: 10.1007/s00208-006-0069-8
URI: http://eprints.whiterose.ac.uk/id/eprint/7001

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